Lesson 2: Sum of Angles in a Polygon
Length of Lesson: 50-70 minutes
Intermath Task
What is the sum of the angles of a triangle? What is the sum of the angles in a
quadrilateral? What is the sum of the angles in a pentagon? What is the sum of the angles
in a hexagon? What is the sum of the angles in convex polygons in terms of the number
of sides?
Essential Questions (rationale):
1. What are quadrilateral, pentagon, and hexagon?
2. Can all of the above figures be classified as polygons?
3. What are angles?
4. How would the sum of the interior angles of a polygon be found?
Intermath Alignment:
This lesson provides students with an investigation of polygons. It also engages students
in a hand on approach to the exploration. Students should gain a stronger understanding
through the use of concrete representation. This approach will provide the scaffolding
need until they are able to understand the more abstract ideas presented in textbooks and
worksheets.
Objectives:
Students will be able to work in groups to calculate the sum of the interior angles of a
particular polygon.
Students will be able to collect data from other groups and find patterns among the
information.
Students will be able to develop a formula for the sum of the interior angles of an n-gon
and test that formula for several cases.
Students will be able to prove the formula by an alternative method.
Procedures:
1. Make 21 slips of paper with three each of the numbers 4-10. This will create seven
groups of three.
2. Once students are in their groups of three, they should notice that each has either a red
dot (for spokesperson), a blue dot (for recorder), or a green dot (for researcher). The
number on their cards indicates the number of sides on their polygon.
3. The spokesperson from each group may go to the supply table and get the following
items for the group: a construction paper cutout of their designated polygon, a ruler, and a
chart with the headings "# of sides", "# of triangles", and "sum of interior angles".
4. Working together in their groups, students must figure out the sum of the interior
angles of their polygon. They may not use a protractor. The students may use triangles
cut out, but their main tool should be their previous knowledge of the sum of the interior
angles of a triangle. (180)
5. Ideally, students will use their ruler to connect one vertex of their polygon to all other
vertices, thus creating triangles within their polygon.
6. Once the secretary has recorded their answers on the chart as well as the method used
to obtain those answers (on the back of the worksheet), the researcher should visit the
other groups to collect their data.
7. After the groups' charts are filled out with information about each polygon (sides 410), the students should look carefully for patterns in their data.
8. The teacher may suggest that "# of sides" be named "n" and the each of the other
categories be written in terms of n.
9. Ultimately, each group should come up with the formula (n - 2)*180 for the sum of the
interior angles of an n-gon. At the end of the lesson, each spokesperson should report the
formula found by their group. In order to check the accuracy of the formula agreed upon
by the whole class, all seven spokespeople should come to the board and prove that the
formula works for their polygon. After the presentation from each group, the students
will use GSP to check their findings.
Assessment:
Informal—teacher will monitor and observe progress; teacher will assist as needed;
teacher will also give cue or clues to help students to remember prior lessons.
Formal-through the spokesperson's presentation at the board, he/she can demonstrate the
group's understanding of the lesson. The secretary should turn in the group’s chart to the
teacher. For homework, students should attempt to write a formal proof using a
"Statements and Reasons" chart to show that the sum of the measures of the interior
angles of an n-gon is (n - 2)*180.
Reflection:
To show the need for a “short cut” or formula, I would start each group with three
figures, 14, 15, and 20-sided figures. I would ask each group choose a vertex and draw a
diagonal to all of the non-adjacent vertices in the figure. Hopefully students will see that
as the number of sides increase so does the number of triangles created by the diagonal.
I hope this will lead them to think that a formula or a short cut is necessary.
Task Appropriateness:
The original Intermath assignment has been modified in order to accommodate all levels
of readiness. The main focus in grade six is the basic construction and properties of
polygons. The students were provided with each polygon and a table to record their
findings because the focus is the sum of the interior angles not constructing the polygons.
Objective Alignment:
M5M1Students will extend their understanding of area of fundamental geometric
plane figures.
a. Find the areas of triangles and parallelograms using formulae.
b. Find the area of a polygon (regular and irregular) by dividing it into squares,
rectangles, and/or triangles and finding the sum of the areas of those shapes.
Multiple Representation: For additional remediation students will use Geoboards and
colored rubber bands to construct as many triangles as possible for a given polygon. This
hands-on activity provides a visual representation for the visual learners. This activity
will also enhance understand for the tactile learner.
Questions to Anticipate:
Will all the vertices be use?
How many polygons will be investigated?
What is a polygon?
Can I make a polygon from triangles?